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In mathematics, a product is the result of multiplication, or an expression that identifies factors to be multiplied. For example, 30 is the product of 6 and 5 (the result of multiplication), and is the product of and (indicating that the two factors should be multiplied together).
The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, is non-commutative, and so is multiplication in other algebras in general as well.
There are many different kinds of products in mathematics: besides being able to multiply just numbers, polynomials or matrices, one can also define products on many different algebraic structures.
Product of two numbers
Product of two natural numbers
3 by 4 is 12
Placing several stones into a rectangular pattern with rows and columns gives
Product of two integers
Integers allow positive and negative numbers. Their product is determined by the product of their positive amounts, combined with the sign derived from the following rule:
(This rule is a necessary consequence of demanding distributivity of multiplication over addition, and is not an additional rule.)
In words, we have:
Minus times Minus gives Plus
Minus times Plus gives Minus
Plus times Minus gives Minus
Plus times Plus gives Plus
Product of two fractions
Two fractions can be multiplied by multiplying their numerators and denominators:
The geometric meaning is that the magnitudes are multiplied and the arguments are added.
Product of two quaternions
The product of two quaternions can be found in the article on quaternions. Note, in this case, that and are in general different.
Product of a sequence
The product operator for the product of a sequence is denoted by the capital Greek letter pi? (in analogy to the use of the capital Sigma ? as summation symbol). For example, the expression is another way of writing .
The product of a sequence consisting of only one number is just that number itself; the product of no factors at all is known as the empty product, and is equal to 1.
The product of two polynomials is given by the following:
Products in linear algebra
There are many different kinds of products in linear algebra. Some of these have confusingly similar names (outer product, exterior product) with very different meanings, while others have very different names (outer product, tensor product, Kronecker product) and yet convey essentially the same idea. A brief overview of these is given in the following sections.
By the very definition of a vector space, one can form the product of any scalar with any vector, giving a map .
Now we consider the composition of two linear mappings between finite dimensional vector spaces. Let the linear mapping f map V to W, and let the linear mapping g map W to U. Then one can get
Or in matrix form:
in which the i-row, j-column element of F, denoted by Fij, is fji, and Gij=gji.
The composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication.
Product of two matrices
Given two matrices
their product is given by
Composition of linear functions as matrix product
There is a relationship between the composition of linear functions and the product of two matrices. To see this, let r = dim(U), s = dim(V) and t = dim(W) be the (finite) dimensions of vector spaces U, V and W. Let
be a basis of U,
be a basis of V and
be a basis of W. In terms of this basis, let
be the matrix representing f : U -> V and
be the matrix representing g : V -> W. Then
is the matrix representing .
In other words: the matrix product is the description in coordinates of the composition of linear functions.
Tensor product of vector spaces
Given two finite dimensional vector spaces V and W, the tensor product of them can be defined as a (2,0)-tensor satisfying:
The tensor product, outer product and Kronecker product all convey the same general idea. The differences between these are that the Kronecker product is just a tensor product of matrices, with respect to a previously-fixed basis, whereas the tensor product is usually given in its intrinsic definition. The outer product is simply the Kronecker product, limited to vectors (instead of matrices).
The class of all objects with a tensor product
In general, whenever one has two mathematical objects that can be combined in a way that behaves like a linear algebra tensor product, then this can be most generally understood as the internal product of a monoidal category. That is, the monoidal category captures precisely the meaning of a tensor product; it captures exactly the notion of why it is that tensor products behave the way they do. More precisely, a monoidal category is the class of all things (of a given type) that have a tensor product.
Other products in linear algebra
Other kinds of products in linear algebra include:
All of the previous examples are special cases or examples of the general notion of a product. For the general treatment of the concept of a product, see product (category theory), which describes how to combine two objects of some kind to create an object, possibly of a different kind. But also, in category theory, one has:
A function's product integral (as a continuous equivalent to the product of a sequence or as the multiplicative version of the normal/standard/additive integral. The product integral is also known as "continuous product" or "multiplical".